11n
81
(K11n
81
)
1
Arc Sequences
4 1 9 2 11 1 10 4 7 8 6
Solving Sequence
1,4 2,7,9
10 3 6 8 11 5
c
1
c
9
c
3
c
6
c
8
c
11
c
5
c
2
, c
4
, c
7
, c
10
Representation Ideals
I =
7
\
i=1
I
u
i
\
I
v
1
I
u
1
= hc, u 1, b + 1, da 1i
I
u
2
= ha, c, u 1, d + 1, b + 1i
I
u
3
= hb, c, u 1, a + 1, d + 1i
I
u
4
= hb u, u
3
+ 2u
2
+ a 1, u
4
2u
2
+ d + 2u, u
4
u
3
+ 2u
2
+ c 2, u
5
+ 2u
4
2u
3
3u
2
+ 3u + 1i
I
u
5
= hb u, u
4
u
3
+ 3u
2
+ 2a + 5u, u
3
u
2
+ 2d 3u + 1, u
5
+ u
4
4u
3
4u
2
+ 3u 1,
u
4
2u
3
+ 4u
2
+ 2c + 8u 1i
I
u
6
= h−u
3
+ a + 2u 2, u
4
2u
2
+ d + 2u, u
4
u
3
+ 2u
2
+ c 2, u
5
+ 2u
4
2u
3
3u
2
+ 3u + 1,
u
4
u
3
+ 2u
2
+ b + u 1i
I
u
7
= hu
5
u
3
+ 3u
2
4, u
3
+ 2a u + 1, u
4
2u
3
u
2
+ 4b + 5u 2, 3u
4
+ 2u
3
u
2
+ 8c 3u + 10,
5u
4
6u
3
+ 3u
2
+ 4d + 9u 14i
I
v
1
= ha, d, b 1, v 1, c + 1i
There are 8 irreducible components with 23 representations.
There are 1 irreducible components of dim
C
= 1 for 11n
81
1
The knot diagram image is adapter from “C. Livingston and A. H. Moore, KnotInfo: Table of Knot
Invariants, http://www.indiana.edu/ knotinfo”
1
I. I
u
1
= hc, u 1, b + 1, da 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
7
=
a
1
a
9
=
0
d
a
10
=
a
d + 1
a
3
=
0
1
a
6
=
a 1
1
a
8
=
0
d
a
11
=
a
1
a
5
=
1
0
a
5
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = ···
a = ···
b = ···
c = ···
d = ···
4.93480 17.8199 + 0.3614I
3
II. I
u
2
= ha, c, u 1, d + 1, b + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
7
=
0
1
a
9
=
0
1
a
10
=
0
1
a
3
=
0
1
a
6
=
1
1
a
8
=
0
1
a
11
=
0
1
a
5
=
1
0
a
5
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0
b = 1.00000
c = 0
d = 1.00000
3.28987 12.0000
5
III. I
u
3
= hb, c, u 1, a + 1, d + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
7
=
1
0
a
9
=
0
1
a
10
=
1
1
a
3
=
0
1
a
6
=
1
0
a
8
=
0
1
a
11
=
1
0
a
5
=
1
0
a
5
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0
c = 0
d = 1.00000
3.28987 12.0000
7
IV. I
u
4
= hb u, u
3
+ 2u
2
+ a 1, u
4
2u
2
+ d + 2u, u
4
u
3
+ 2u
2
+ c
2, u
5
+ 2u
4
+ · · · + 3u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
7
=
u
3
2u
2
+ 1
u
a
9
=
u
4
+ u
3
2u
2
+ 2
u
4
+ 2u
2
2u
a
10
=
u
3
2u
2
+ 1
1
a
3
=
u
2
+ 1
u
2
a
6
=
u
3
2u
2
+ u + 1
u
a
8
=
u
4
+ u
3
2u
2
+ 2
u
4
+ u
3
2u
2
+ 1
a
11
=
u
4
+ 2u
3
u
2
u + 1
u
2
a
5
=
u
u
3
+ u
a
5
=
u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.81245 0.17314I
a = 0.280849 + 0.445862I
b = 1.81245 0.17314I
c = 0.099771 + 1.129447I
d = 0.06615 2.48427I
15.1998 4.1249I 13.10604 + 2.15443I
u = 1.81245 + 0.17314I
a = 0.280849 0.445862I
b = 1.81245 + 0.17314I
c = 0.099771 1.129447I
d = 0.06615 + 2.48427I
15.1998 + 4.1249I 13.10604 2.15443I
u = 0.274898
a = 0.869636
b = 0.274898
c = 1.83380
d = 0.695222
2.08622 3.05696
u = 0.949895 0.441667I
a = 0.71567 + 2.78754I
b = 0.949895 0.441667I
c = 0.682871 0.618084I
d = 0.281458 + 0.392024I
5.14125 + 1.10891I 14.3655 2.0411I
u = 0.949895 + 0.441667I
a = 0.71567 2.78754I
b = 0.949895 + 0.441667I
c = 0.682871 + 0.618084I
d = 0.281458 0.392024I
5.14125 1.10891I 14.3655 + 2.0411I
9
V. I
u
5
= hb u, u
4
u
3
+ 3u
2
+ 2a + 5u, u
3
u
2
+ 2d 3u + 1, u
5
+ u
4
+
· · · + 3u 1, u
4
2u
3
+ · · · + 2c 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
7
=
1
2
u
4
+
1
2
u
3
3
2
u
2
5
2
u
u
a
9
=
1
2
u
4
+ u
3
2u
2
4u +
1
2
1
2
u
3
+
1
2
u
2
+
3
2
u
1
2
a
10
=
1
2
u
4
+
1
2
u
3
3
2
u
2
5
2
u
1
2
u
3
1
2
u
2
+
3
2
u
1
2
a
3
=
u
2
+ 1
u
2
a
6
=
1
2
u
4
+
1
2
u
3
3
2
u
2
3
2
u
u
a
8
=
1
2
u
4
+ u
3
2u
2
4u +
1
2
1
2
u
4
1
2
u
3
+
3
2
u
2
+
1
2
u
a
11
=
1
2
u
3
1
2
u
2
+
3
2
u +
1
2
u
2
a
5
=
u
u
3
+ u
a
5
=
u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 1.72935 0.51571I
a = 0.461516 + 1.228873I
b = 1.72935 0.51571I
c = 0.297131 1.134289I
d = 0.16439 + 2.36316I
16.6614 10.9560I 13.7735 + 4.2698I
u = 1.72935 + 0.51571I
a = 0.461516 1.228873I
b = 1.72935 + 0.51571I
c = 0.297131 + 1.134289I
d = 0.16439 2.36316I
16.6614 + 10.9560I 13.7735 4.2698I
u = 0.287923 0.283171I
a = 0.759868 + 0.928220I
b = 0.287923 0.283171I
c = 0.71581 + 1.41065I
d = 0.044061 0.482429I
0.341586 + 0.921914I 6.28644 7.57142I
u = 0.287923 + 0.283171I
a = 0.759868 0.928220I
b = 0.287923 + 0.283171I
c = 0.71581 1.41065I
d = 0.044061 + 0.482429I
0.341586 0.921914I 6.28644 + 7.57142I
u = 1.88286
a = 0.403297
b = 1.88286
c = 1.16265
d = 0.759351
17.8353 13.8801
11
VI. I
u
6
= h−u
3
+ a + 2u 2, u
4
2u
2
+ d + 2u, u
4
u
3
+ 2u
2
+ c
2, u
5
+ 2u
4
+ · · · + 3u + 1, u
4
u
3
+ · · · + b 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
7
=
u
3
2u + 2
u
4
+ u
3
2u
2
u + 1
a
9
=
u
4
+ u
3
2u
2
+ 2
u
4
+ 2u
2
2u
a
10
=
u
3
2u + 2
1
a
3
=
u
2
+ 1
u
2
a
6
=
u
4
+ 2u
3
2u
2
3u + 3
u
4
+ u
3
2u
2
u + 1
a
8
=
u
4
+ u
3
2u
2
+ 2
u
4
+ u
3
2u
2
+ 1
a
11
=
u
4
u
3
+ 3u
2
+ u 3
u
4
+ 3u
2
2u 2
a
5
=
u
u
3
+ u
a
5
=
u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 1.81245 0.17314I
a = 0.165924 1.354819I
b = 0.71268 + 1.30259I
c = 0.099771 + 1.129447I
d = 0.06615 2.48427I
15.1998 4.1249I 13.10604 + 2.15443I
u = 1.81245 + 0.17314I
a = 0.165924 + 1.354819I
b = 0.71268 1.30259I
c = 0.099771 1.129447I
d = 0.06615 + 2.48427I
15.1998 + 4.1249I 13.10604 2.15443I
u = 0.274898
a = 2.52902
b = 1.10870
c = 1.83380
d = 0.695222
2.08622 3.05696
u = 0.949895 0.441667I
a = 0.401414 0.226060I
b = 1.267024 0.176417I
c = 0.682871 0.618084I
d = 0.281458 + 0.392024I
5.14125 + 1.10891I 14.3655 2.0411I
u = 0.949895 + 0.441667I
a = 0.401414 + 0.226060I
b = 1.267024 + 0.176417I
c = 0.682871 + 0.618084I
d = 0.281458 0.392024I
5.14125 1.10891I 14.3655 + 2.0411I
13
VII. I
u
7
= hu
5
u
3
+ 3u
2
4, u
3
+ 2a u + 1, u
4
2u
3
+ · · · + 4b
2, 3u
4
+ 2u
3
+ · · · + 8c + 10, 5u
4
6u
3
+ · · · + 4d 14i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
7
=
1
2
u
3
+
1
2
u
1
2
1
4
u
4
+
1
2
u
3
+ ···
5
4
u +
1
2
a
9
=
3
8
u
4
1
4
u
3
+ ··· +
3
8
u
5
4
5
4
u
4
+
3
2
u
3
+ ···
9
4
u +
7
2
a
10
=
1
2
u
3
+
1
2
u
1
2
3
4
u
4
+
3
2
u
3
+ ···
7
4
u +
3
2
a
3
=
u
2
+ 1
u
2
a
6
=
1
4
u
4
+
1
4
u
2
3
4
u
1
4
u
4
+
1
2
u
3
+ ···
5
4
u +
1
2
a
8
=
3
8
u
4
1
4
u
3
+ ··· +
3
8
u
5
4
3
4
u
4
+
1
2
u
3
+ ···
3
4
u +
5
2
a
11
=
1
8
u
4
1
4
u
3
+ ··· +
5
8
u
1
4
1
2
u
4
+
1
2
u
2
+
1
2
u 2
a
5
=
u
u
3
+ u
a
5
=
u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
14
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
7
1(vol +
1CS) Cusp shape
u = 1.267024 0.176417I
a = 0.175654 + 0.333863I
b = 0.949895 0.441667I
c = 0.195051 + 0.728580I
d = 0.55136 2.96396I
5.14125 + 1.10891I 14.3655 2.0411I
u = 1.267024 + 0.176417I
a = 0.175654 0.333863I
b = 0.949895 + 0.441667I
c = 0.195051 0.728580I
d = 0.55136 + 2.96396I
5.14125 1.10891I 14.3655 + 2.0411I
u = 0.71268 1.30259I
a = 1.48919 0.76398I
b = 1.81245 + 0.17314I
c = 1.077607 + 0.878534I
d = 0.431826 0.856727I
15.1998 + 4.1249I 13.10604 2.15443I
u = 0.71268 + 1.30259I
a = 1.48919 + 0.76398I
b = 1.81245 0.17314I
c = 1.077607 0.878534I
d = 0.431826 + 0.856727I
15.1998 4.1249I 13.10604 + 2.15443I
u = 1.10870
a = 0.627062
b = 0.274898
c = 0.454684
d = 0.239061
2.08622 3.05696
15
VIII. I
v
1
= ha, d, b 1, v 1, c + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
1
0
a
2
=
1
0
a
7
=
0
1
a
9
=
1
0
a
10
=
1
1
a
3
=
1
0
a
6
=
1
1
a
8
=
1
0
a
11
=
0
1
a
5
=
1
0
a
5
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
16
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
c = 1.00000
d = 0
3.28987 12.0000
17
IX. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
u(u 1)
2
(u
5
u
3
+ 3u
2
4)(u
5
+ u
4
4u
3
4u
2
+ 3u 1)
(u
5
+ 2u
4
2u
3
3u
2
+ 3u + 1)
2
c
2
u(u + 1)
2
(u
5
+ 2u
4
+ u
3
+ 9u
2
+ 24u + 16)
(1 + 15u + 25u
2
+ 22u
3
+ 8u
4
+ u
5
)
2
(u
5
+ 9u
4
+ ··· + u + 1)
c
3
, c
8
u
3
(u
5
u
4
+ 5u
3
u
2
+ 2u + 2)
3
(u
5
+ 4u
4
+ 8u
3
+ 8u
2
+ 4)
c
4
, c
5
, c
6
c
9
, c
10
, c
11
u(u 1)(u + 1)(u
5
u
3
+ 3u
2
4)(u
5
+ u
4
4u
3
4u
2
+ 3u 1)
(u
5
+ 2u
4
2u
3
3u
2
+ 3u + 1)
2
18
X. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
5
c
6
, c
7
, c
9
c
10
, c
11
y(y 1)
2
(y
5
9y
4
+ 30y
3
38y
2
+ y 1)
(1 + 15y 25y
2
+ 22y
3
8y
4
+ y
5
)
2
(y
5
2y
4
+ ··· + 24y 16)
c
2
y(y 1)
2
(y
5
21y
4
+ 218y
3
1402y
2
75y 1)
(y
5
20y
4
+ 114y
3
+ 19y
2
+ 175y 1)
2
(y
5
2y
4
+ 13y
3
97y
2
+ 288y 256)
c
3
, c
8
y
3
(y
5
96y
2
64y 16)(y
5
+ 9y
4
+ 27y
3
+ 23y
2
+ 8y 4)
3
19